Classical Signal Detection
The problem of detecting the presence of signals in noise is a classical problem in statistical signal processing. In the case where the transmission of information is not a concern, the detection problem is then typically that of testing the presence or absence of a signal or signals. This detection problem may be formulated as a binary hypothesis test. That is, a decision is made between two hypotheses, H0 and H1 defined as follows:                H0: only noise is present,        H1: a signal or signals plus noise is present.Detection is based on a decision statistic, C, that is some function of the received signal which is compared to a threshold, τ. If the threshold is exceeded, H1 is decided to be true, otherwise H0 is true. This decision rule can be expressed as follows:        
  C  ⁢            ≷      _              H      0              H      1        ⁢  τThe performance of the detector is often characterized in terms of the probability of detection Pd and probability of false alarm PFA. The probability of detection is the probability that H1 is selected given that H1 is true, i.e., Pd=Pr{C>τ|H1}. The probability of false alarm is the probability that H1 is selected given that H0 is true, i.e., PFA=Pr{C>τ|H0}. In either case, the probability distribution of the decision statistic must be defined to specify either Pd or PFA.
The transmitted signals can be known, but with unknown parameters, or can be completely unknown. Hence, for a given decision statistic, accurately characterizing the probability of detection can be difficult, if not impossible, in certain realistic propagation environments. An alternative is to limit the chances of declaring a signal is present when no signal is present, i.e., appropriately setting the detection threshold for a given probability of false alarm. Setting the threshold requires knowledge of the probability distribution of the decision statistic under H0. A processor wherein the threshold is adjusted to maintain a certain PFA falls under the class of constant false alarm rate, CFAR, detectors.
Signal Detection Using the FFT
A common decision statistic that is used to determine if man-made signals are present in a particular portion of the frequency spectrum is the magnitude squared of the discrete Fourier transform (DFT) outputs of a sampled input time series from the environment. Let x[n] be a length N sampled input of measurements within some frequency band. In general, x[n] are complex values. The discrete Fourier transform of x[n] may be defined as
      X    ⁡          [      k      ]        =            ∑              n        =        0                    N        -        1              ⁢                  x        ⁡                  [          n          ]                    ⁢              exp        ⁡                  (                                    -              j                        ⁢                                          2                ⁢                π                            N                        ⁢            nk                    )                    where k=0, . . . ,N−1 and j=√{square root over (−1)}. In practice, the DFT may be computed efficiently with the Fast Fourier Transform (FFT) algorithm. To detect man-made energy in the k-th FFT bin, one may use the following decision rule and statistic
                          X        ⁡                  [          k          ]                            2    ⁢            ≷      _              H      0              H      1        ⁢      τ    .  
Typically, x[n] is multiplied by a window function w[n] of length N so that one can distinguish energy in different bands. The most basic window function w[n] is where all the elements are equal to 1 and is commonly referred to as the rectangular window. Therefore, the discrete Fourier transform of the windowed sequence is
      X    ⁡          [      k      ]        =            ∑              n        =        0                    N        -        1              ⁢                  x        ⁡                  [          n          ]                    ⁢              w        ⁡                  [          n          ]                    ⁢                        exp          ⁡                      (                                          -                j                            ⁢                                                2                  ⁢                  π                                N                            ⁢              nk                        )                          .            
Depending on the application and detection strategy, multiple X[k]'s can be used in a decision rule. An example is outlined in R. Inkol, S. Wang and F. Patenaude, “Upper and Lower Bounds for the Threshold of the FFT Filter Bank-based Summation CFAR Detector,” ICASSP 2006 Proceedings, Vol 3, pp 289-292, May 2006.
Problems With Prior Art
Many conventional detectors are deficient in that their detection functionality is dependent upon having an accurate estimate of the noise power. For example, some conventional detectors, under certain environments where the signal-to-noise power ratio can change abruptly, e.g., wireless channels, cannot change their detection threshold without having to restart their numerical algorithm to estimate the noise power.
The FFT-based detection procedure described in U.S. Pat. No. 5,323,337 to Wilson et al. entitled “Signal Detector Employing Mean Energy and Variance of Energy Content Comparison for Noise Detection” detects noise at certain FFT bins. However, a problem with this procedure is that it does not provide quantitative values for the noise detection threshold. Thus, the Wilson et al. detection procedure does not guarantee that a certain false alarm rate will be achieved.
Two disadvantages of prior techniques that use the FFT for detecting man-made energy within frequency bands are: 1) the computationally intensive nature of the method for determining the threshold, the method also requiring a calibration period to estimate the statistics of the environment, namely the noise variance, and 2) the inability to immediately adapt to abrupt changes in the environment such as signal and/or noise power fluctuations, which create unexpected and oftentimes higher false alarm rates.
It would be useful to be able to provide a signal detection technology that overcomes or mitigates one or more of the disadvantages of prior FFT-signal detection techniques.